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%%文档的题目、作者与日期
%\author{王立庆（2022级数学与应用数学1班）}
\author{ALEX }
\title{高等代数复习题 - 向量空间}
%\date{\vspace{-3ex}}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{2022 年 9 月 8 日}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %1
已知 $\mathbb{R}^n$ 中的任意向量均可由向量组 $\alpha_1, \cdots, \alpha_n$ 线性表示，则 $\alpha_1, \cdots, \alpha_n$ 具有下述哪个性质？
\begin{enumerate}
\item  线性相关。
\item  秩为 $n$.
\item  秩小于 $n$.
\item  以上都不对。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：(b). 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %2
设 $A$ 为 $n$ 阶方阵且 $|A|=0$, 则下述说法中，正确的是哪个？

\begin{enumerate}
\item  $A$ 中必有两行(列) 元素对应成比例。
\item  $A$ 中至少有一行(列) 元素为零。
\item  $A$ 中至少有一行元素是其余各行向量的线性组合。
\item  $A$ 中每一行向量都是其余各行向量的线性组合。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：(c). 
%
%}

\vspace{0.2cm}


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\item  %3
关于一个向量组的极大线性无关组，下述说法中，正确的是哪个？

\begin{enumerate}
\item  极大线性无关组的个数是唯一的。
\item  极大线性无关组的个数是不唯一的。
\item  所有极大线性无关组所含向量个数是一样的。
\item  所有极大线性无关组所含向量个数可能是不一样的。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：(c). 
%
%}

\vspace{0.2cm}

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\item  %4
设 $n>1$, 设向量组 $\alpha_1, \cdots, \alpha_n$ 线性相关，其中向量 $\alpha_1$ 不是零向量。
下述说法中，正确的是哪个？

\begin{enumerate}
\item  每个向量 $\alpha_i \ (i>1)$ 都能由 $\alpha_1, \cdots, \alpha_{i-1}$ 线性表示。
\item  每个向量 $\alpha_i \ (i>1)$ 都不能由 $\alpha_1, \cdots, \alpha_{i-1}$ 线性表示。
\item  某一个向量 $\alpha_i \ (i>1)$ 能由 $\alpha_1, \cdots, \alpha_{i-1}$ 线性表示。
\item  某一个向量 $\alpha_i \ (i>1)$ 不能由 $\alpha_1, \cdots, \alpha_{i-1}$ 线性表示。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：(c). 
%
%}

\vspace{0.2cm}


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\item  %5
设 $m\times n$ 矩阵 $A$ 中的 $n$ 个列向量线性无关。下述说法中，正确的是哪个？

\begin{enumerate}
\item  $R(A)$ 大于 $m$.
\item  $R(A)$ 大于 $n$.
\item  $R(A)$ 等于 $m$.
\item  $R(A)$ 等于 $n$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：(b). 
%
%}

\vspace{0.2cm}


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\item %6
设 $A$ 是 $n$ 阶方阵, 设 $R(A)=r<n$. 
下述说法中，正确的是哪个？

\begin{enumerate}
\item  在 $A$ 的行向量中必有 $r$ 个行向量线性无关。
\item  在 $A$ 的行向量中任意 $r$ 个行向量线性无关。
\item  在 $A$ 的行向量中任意 $r$ 个行向量都构成极大线性无关组。
\item  在 $A$ 的行向量中任意一个行向量都可以由其他 $r$ 个行向量线性表示。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：(a). 
%
%}

\vspace{0.2cm}


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\item %7
设 $A$ 是 $n$ 阶方阵，下述哪个是 $R(A)=r<n$ 的充要条件？

\begin{enumerate}
\item  $A$ 的任意一个 $r$ 阶子式都不等于零。
\item  $A$ 的任意一个 $r+1$ 阶子式都不等于零。
\item  $A$ 的任意 $r$ 个列向量线性无关。
\item  $A$ 的任意 $r+1$ 个列向量线性相关，而有 $r$ 个列向量线性无关。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：(d).
%
%}

\vspace{0.2cm}

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\item %8
设 $P$ 是一个数域，证明 $P^{m\times n}$ 是 $P$ 上的线性空间，而且 $\dim P^{m\times n}=mn$. 

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：记 $E_{ij}$ 是第 $(i, j)$ 元素为 1, 其余元素为 0 的 $m\times n$ 矩阵。
%证明 $\{ E_{ij} \mid 1\leq i\leq m, 1\leq j\leq n\}$ 为 $P^{m\times n}$ 的一个基。
%
%}

\vspace{0.2cm}

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\item  %9
设 $P$ 是一个数域，证明 $P^{m}$ 是 $P$ 上的线性空间，且 $\dim P^{m}=m$. 

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%证明 $\{ \varepsilon_i = (0, \cdots, 1, \cdots, 0) \mid 1\leq i\leq m\}$ 是一个基。
%
%}

\vspace{0.2cm}

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\item  %10
求向量 $\xi = (a_1, a_2)\in R^2$ 在 $R^2$ 的标准基 
$\{ \varepsilon_1=(1,0), \varepsilon_2=(0,1) \}$ 下的坐标。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：坐标为 $(a_1, a_2)$.
%
%}

\vspace{0.2cm}

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\item  %11
求向量 $\xi = (a_1, a_2)\in R^2$ 在 $R^2$ 的一个基 
$\{\alpha_1=(2,1), \alpha_2=(1,1)\}$ 下的坐标。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：坐标为 $(a_1-a_2, 2a_2-a_1)$.
%
%}

\vspace{0.2cm}

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\item  %12
求次数小于等于 $n$ 的实系数多项式全体 $\mathbb{R}[x]_{n}$ 的一个基和它的维数。 

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：一个基为 $(1,x,x^2,\cdots,x^n)$, 维数为 $n+1$. 
%
%}

\vspace{0.2cm}

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\item  %13
求次数小于等于 $n$ 的实系数多项式全体 $\mathbb{R}[x]_{n}$ 中任意多项式在基 $(1,x,x^2,\cdots,x^n)$ 下的坐标。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：多项式 $f(x)=a_0+a_1x+\cdots+a_nx^n$ 在基 $(1,x,x^2,\cdots,x^n)$ 下的坐标为 
%$(a_0,a_1,\cdots,a_n)$. 
%
%}

\vspace{0.2cm}

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\item  %14
求 $P^{2\times 2}$ 的一个基，和维数。 
求 $P^{2\times 2}$ 中任意矩阵在这组基下的坐标。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：一个基可以取为 
%$%\begin{eqnarray*}
%E_{11}=\begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}, \,\,
%E_{12}=\begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix}, \,\,
%E_{21}=\begin{bmatrix} 0&0 \\ 1&0 \end{bmatrix}, \,\,
%E_{22}=\begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}. 
%$%\end{eqnarray*}
%任意矩阵
%$%\begin{eqnarray*}
%E_{11}=\begin{bmatrix} a&b \\ c&d \end{bmatrix}
%$%\end{eqnarray*}
%在这个基下的坐标为 $(a,b,c,d)$. 
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %15
在 $P^4$ 中, 有两个基
\begin{eqnarray*}
\Phi &=& (\varepsilon_1=(1,1,1,1), \varepsilon_1=(1,1,-1,-1), \varepsilon_3=(1,-1,1,-1), \varepsilon_4=(1,-1,-1,1)), \\
\Psi &=& (\eta_1=(1,1,0,1), \eta_1=(1,0,0,0), \eta_3=(0,1,0,0), \eta_4=(0,0,1,0)). 
\end{eqnarray*}
求 $\alpha=(0,0,0,1)$ 分别在这两个基下的坐标，这两个坐标有什么关系吗?

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：根据坐标的定义，求坐标就是解线性方程组 
%\begin{eqnarray*}
%x_1\varepsilon_1 + x_2\varepsilon_2 + x_3\varepsilon_3 + x_4\varepsilon_4 &=& \alpha, \\
%y_1\eta_1 + y_2\eta_2 + y_3\eta_3 + y_4\eta_4 &=& \alpha. 
%\end{eqnarray*}
%写成列向量的形式，得到 $\Phi$ 与 $\Psi$ 为 $4\times 4$ 矩阵。把坐标也写成列向量的形式， $x=(x_1,x_2,x_3,x_4)^t$, $y=(y_1,y_2,y_3,y_4)^t$, 则有 $\Phi\cdot x = \Psi\cdot y$. 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %16
求全体复数的集合 $\mathbb{C}$ 看成复数域 $\mathbb{C}$ 上的向量空间的维数与一组基，并求出任一元素在这个基下的坐标。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：任意一个非零复数都是这个一维向量空间的基。取基为 $\varepsilon=1$, 则任意复数 $z$ 的坐标是 $z$. 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %17
求全体复数的集合 $\mathbb{C}$ 看成实数域 $\mathbb{R}$ 上的向量空间的维数与一组基。并求出任一元素在这个基下的坐标。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：任意两个不成比例的复数都是这个二维向量空间的基。
%取基为 $\alpha_1=1, \alpha_2=i$. 则任意复数 $z=x+yi$ 的坐标是 $(x,y)$. 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %18
设 $\alpha_1, \alpha_2, \alpha_3$ 是 3 维向量空间 $\mathbb{R}^3$ 的一个基。
则从基 $\Phi=(\alpha_1, \frac{1}{2}\alpha_2, \frac{1}{3}\alpha_3)$ 到
另一个基 $\Psi = (\alpha_1+\alpha_2, \alpha_2+\alpha_3, \alpha_3+\alpha_1)$ 的过渡矩阵是什么？

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：按定义，所求过渡矩阵是下述等式右边的三阶矩阵，
%\begin{eqnarray*}
%(\alpha_1+\alpha_2, \,\, \alpha_2+\alpha_3, \,\, \alpha_3+\alpha_1) = 
%(\alpha_1, \,\, \frac{1}{2}\alpha_2, \,\, \frac{1}{3}\alpha_3)
%\begin{bmatrix} 1&0&1 \\ 2&2&0 \\ 0&3&3 \end{bmatrix}. 
%\end{eqnarray*}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %19
设 $(\alpha_1, \cdots, \alpha_n)$ 与 $(\beta_1, \cdots, \beta_n)$ 为向量空间 $\mathbb{R}^n$ 的两个基，设 $\alpha\in \mathbb{R}^n$, 
设 $A, B$ 是 $n$ 阶矩阵，设有
\begin{eqnarray*}
(\alpha_1, \cdots, \alpha_n) &=& (\beta_1, \cdots, \beta_n)A, \\ 
\alpha=x_1\alpha_1+\cdots+x_n\alpha_n &=& y_1\beta_1+\cdots+y_n\beta_n, \\ 
(x_1, \cdots, x_n) &=& (y_1, \cdots, y_n)B. 
\end{eqnarray*}
则 $A, B$ 之间有什么关系?


%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：$AB=E$. 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %20
设 $P^4$ 中的两个基如下，求一非零向量 $\xi$, 使得它在这两组基下的坐标相同，
\begin{eqnarray*}
\left\{ \begin{array}{rcl}
\varepsilon_1 &=& (1,0,0,0), \\
\varepsilon_2 &=& (0,1,0,0), \\
\varepsilon_3 &=& (0,0,1,0), \\
\varepsilon_4 &=& (0,0,0,1),
\end{array}\right. 
\left\{ \begin{array}{rcl}
\eta_1 &=& (2,1,-1,1), \\
\eta_2 &=& (0,3,1,0), \\
\eta_3 &=& (5,3,2,1), \\
\eta_4 &=& (6,6,1,3).
\end{array}\right.
\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：设坐标为 $x=(x_1,x_2,x_3,x_4)$. 则问题化为求解线性方程组
%\begin{eqnarray*}
%x_1\varepsilon_1 + x_2\varepsilon_2 + x_3\varepsilon_3 + x_4\varepsilon_4 
%= 
%x_1\eta_1 + x_2\eta_2 + x_3\eta_3 + x_4\eta_4. 
%\end{eqnarray*}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %21
证明：与基础解系等价的线性无关向量组也是基础解系。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：验证基础解系的定义里的几个条件。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %22
设齐次线性方程组 $Ax=0$. 若向量 $\alpha_1, \cdots, \alpha_s \ (s>1)$ 是此线性方程组的基础解系. 证明: $\beta_1=\alpha_2+\cdots+\alpha_s, \beta_2=\alpha_1+\alpha_3+\cdots+\alpha_s, \cdots, \beta_s=\alpha_1+\alpha_2+\cdots+\alpha_{s-1}$ 也是该方程组的基础解系。


%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：验证基础解系的定义里的几个条件。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %23
设 $(\alpha_1, \cdots, \alpha_n)$ 是 $n$ 维向量空间 $V$ 的一个基。
设 $A$ 是一个 $n\times s$ 矩阵，使得 $(\beta_1, \cdots, \beta_s)=(\alpha_1, \cdots, \alpha_n)A$. 
证明: $\dim L(\beta_1, \cdots, \beta_s)=R(A)$.


%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：使用矩阵 $A$ 的相抵标准形。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %24
用基础解系表示齐次方程组的全部解，
\begin{eqnarray*}
\left\{\begin{array}{c}
2x_1+x_2-x_3+x_4=0, \\
x_1+2x_2+x_3-x_4=0, \\
x_1+x_2+2x_3+x_4=0.
\end{array}\right.
\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：将系数矩阵化为行最简形。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %25
判断下述非齐次线性方程组是否有解? 如果有无穷多组解时, 表示全部解。
\begin{eqnarray*}
\left\{\begin{array}{c}
2x_1+x_2-x_3+x_4=1, \\
x_1+2x_2+x_3-x_4=2, \\
x_1+x_2+2x_3+x_4=3. 
\end{array}\right.
\end{eqnarray*}


%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：将增广矩阵化为行最简形。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %26
设 $V_1$ 和 $V_2$ 分别是齐次线性方程组 $x_1+\cdots+x_n=0$ 和 $x_1=\cdots=x_n$ 的解空间。
证明：$\mathbb{R}^n = V_1\oplus V_2$.

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：验证和子空间 $V_1+ V_2 = \mathbb{R}^n$, 以及 $V_1\cap V_2 = \{ \theta \}$. 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %27
令 $M_n(F)$ 是数域 $F$ 上全体 $n$ 阶方阵所组成的向量空间, 令 
\begin{eqnarray*}
S=\{A\in M_n(F)| A^T=A\},\,\, T=\{A\in M_n(F)| A^T=-A\}. 
\end{eqnarray*}
证明: $M_n(F)=S\oplus T$.

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：验证 $S+T = M_n(F)$ 以及 $S\cap T = \{ \theta \}$. 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %28
设矩阵 $A\in \mathbb{R}^{m\times n}$, 且 $A=(A_1,A_2, \cdots, A_n)$, 其中 $A_i$ 为 $A$ 的列向量。
设 $W$ 是 $Ax=0$ 的解空间，设 $U=L(A_1,A_2, \cdots, A_n)$ 为矩阵 $A$ 的列空间。 
证明： $\dim W + \dim U = n$.

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：根据 $\dim U = R(A)$ 即得。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %29
下列集合对于给定的加法和数乘运算是否构成实数域 $\mathbb{R}$ 上的向量空间，并说明理由。

(1) 全体 $n$ 维实向量的集合 $\mathbb{R}^n$, 对于通常的矩阵加法和如下定义的数乘运算 
$$k \circ \alpha = 0, (\forall \alpha \in V, k \in \mathbb{R})$$

(2) 全体 $m\times n$ 实矩阵集合 $\mathbb{R}^{m\times n}$, 对于通常的矩阵加法和如下定义的数乘运算 
$$k \circ A = 0, (\forall A \in V, k \in \mathbb{R})$$

(3) 给定矩阵 $A\in \mathbb{R}^{m\times n}$ 和非零向量 $b\in \mathbb{R}^m$, 非齐次线性方程组 $Ax=b$ 的解向量集合 
$$S=\{x | Ax=b, x\in \mathbb{R}^n\}$$ 对于通常向量的加法和数乘。


%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：否，否，否。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %30
设 $\alpha_1,\alpha_2,\cdots,\alpha_n$ 是 $n$ 维向量空间 $V$ 的一组元素，
且 $V$ 中的任意元素都可用 $\alpha_1,\alpha_2,\cdots,\alpha_n$ 线性表示。
证明 $\alpha_1,\alpha_2,\cdots,\alpha_n$ 是 $V$ 的一个基。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：验证基的定义里的几个条件。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %31
设 $V=\{(a+bi,c+di)\mid a, b, c, d \in \mathbb{R}\}$, 则 $V$ 对于通常的加法和数乘, 求在 $\mathbb{C}$ 上的基和维数? 求在 $\mathbb{R}$ 上的基和维数?

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：分别是2维和4维的。
%
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %32
设 $V_1, V_2$ 是 $V$ 的子空间，且 $V_1\subseteq V_2$. 
证明：如果 $\dim V_1 = \dim V_2$, 那么 $V_1=V_2$.

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：取 $V_1$ 的一个基。证明它也是 $V_2$ 的一个基。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %33
若 $\ell\alpha+m\beta+n\gamma=\theta$ 为零向量且 $\ell n\neq 0$. 证明: $L(\alpha, \beta)=L(\beta, \gamma)$. 

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：验证 $\alpha\in L(\beta, \gamma)$ 以及 $\gamma\in L(\alpha, \beta)$. 
%这样就验证里左边是右边的子集，右边也是左边的子集。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %34
在 $\mathbb{R}^4$ 中，设 $\alpha_1=(2,1,3,1), \alpha_2=(1,2,0,1), \alpha_3=(-1,1,-3,0), \alpha_4=(1,1,1,1)$. 
求 $L(\alpha_1, \alpha_2,\alpha_3, \alpha_4)$ 的基和维数，并将该子空间的基扩成 $\mathbb{R}^4$ 的基。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：将这四个向量按列向量的方式排成一个矩阵，然后用行初等变换将它化为行最简形。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %35
已知两个向量组 
\begin{eqnarray*}
\Phi &=& \{ \alpha_1=(1,3,-2,2,3),  \alpha_2=(1,4,-3,4,2),  \alpha_3=(2,3,-1,-2,9) \}, \\ 
\Psi &=& \{ \beta_1=(1,3,0,2,1),  \beta_2=(1,5,-6,6,3),  \beta_3=(2,5,3,2,1) \} .
\end{eqnarray*}

\begin{enumerate} 
\item  求 $U=L(\alpha_1,\alpha_2,\alpha_3)$ 的一个基和维数。
\item  求 $W=L(\beta_1,\beta_2,\beta_3)$ 的一个基和维数。
\item  求 $U+W$ 的一个基和维数。
\item  求 $U\cap W$ 的一个基和维数。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%\begin{enumerate} 
%\item  将向量 $\alpha_1,\alpha_2,\alpha_3$ 按列向量的方式排成一个矩阵，然后化为行最简形。
%\item  将向量 $\beta_1,\beta_2,\beta_3$ 按列向量的方式排成一个矩阵，然后化为行最简形。
%\item  将向量 $\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2,\beta_3$ 按列向量的方式排成一个矩阵，然后化为行最简形。
%\item  设 $x_1\alpha_1+x_2\alpha_2+x_3\alpha_3 = y_1\beta_1+y_2\beta_2+y_3\beta_3$. 
%将 $(x_1,x_2,x_3,y_1,y_2,y_3)$ 看作未知数，求解这个线性方程组。
%
%\end{enumerate}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %36
已知两个向量组  
\begin{eqnarray*}
\Phi &=& \{ \alpha_1=(1,2,-1,-2), \alpha_2=(3,1,1,1), \alpha_3=(-1,0,1,-1)^T \}, \\ 
\Psi &=& \{ \beta_1=(2,5,-6,-5), \beta_2=(-1,2,-7,3) \}.
\end{eqnarray*}

\begin{enumerate} 
\item  求 $L(\alpha_1, \alpha_2, \alpha_3)+L(\beta_1, \beta_2)$ 的一个基和维数。
\item  求 $L(\alpha_1, \alpha_2, \alpha_3)\cap L(\beta_1, \beta_2)$ 的一个基和维数。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%\begin{enumerate} 
%\item  将向量 $\alpha_1,\alpha_2,\alpha_3,\beta_1,\beta_2$ 按列向量的方式排成一个矩阵，然后化为行最简形。
%\item  设 $x_1\alpha_1+x_2\alpha_2+x_3\alpha_3 = y_1\beta_1+y_2\beta_2$. 
%将 $(x_1,x_2,x_3,y_1,y_2)$ 看作未知数，求解这个线性方程组。
%\end{enumerate}
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %37
已知 $\mathbb{R}^3$ 的两个基 
\begin{eqnarray*}
\Phi &=& \{ \varepsilon_1=(1,1,1), \varepsilon_2=(1,0,-1), \varepsilon_3=(1,0,1) \}, \\ 
\Psi &=& \{ \eta_1=(1,2,1), \eta_2=(2,3,4), \eta_3=(3,4,3) \}.
\end{eqnarray*}

求从 $(\varepsilon_1, \varepsilon_2, \varepsilon_3)$  到 $(\eta_1, \eta_2, \eta_3)$ 的过渡矩阵。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：代入过渡矩阵的定义里的那个等式即得。
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %38
设向量组 $(\alpha_1, \cdots, \alpha_n)$ 是 $n$ 维向量空间 $V$ 的一个基。
\begin{enumerate} 
\item  证明：向量组 $(\alpha_1, \alpha_1+\alpha_2, \cdots, \alpha_1+\cdots+\alpha_n)$ 也是 $V$ 的一个基。
\item  又若 $\alpha$ 关于前一个基的坐标为 $(n, n-1, \cdots, 2, 1)^T$, 求 $\alpha$ 关于后一个基的坐标。
\end{enumerate} 

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%\begin{enumerate} 
%\item  验证基的定义里的几个条件。
%\item  使用过渡矩阵与两个坐标的关系。
%\end{enumerate} 
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %39
已知 $\mathbb{R}^{2\times2}$ 的两个基分别如下，
\begin{eqnarray*}
E_{11} = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}, \,\, 
E_{12} = \begin{bmatrix} 0&1 \\ 0&0 \end{bmatrix}, \,\, 
E_{21} = \begin{bmatrix} 0&0 \\ 1&0 \end{bmatrix}, \,\, 
E_{22} = \begin{bmatrix} 0&0 \\ 0&1 \end{bmatrix}, \\ 
F_{11} = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}, \,\, 
F_{12} = \begin{bmatrix} 1&1 \\ 0&0 \end{bmatrix}, \,\, 
F_{21} = \begin{bmatrix} 1&1 \\ 1&0 \end{bmatrix}, \,\, 
F_{22} = \begin{bmatrix} 1&1 \\ 1&1 \end{bmatrix}.  
\end{eqnarray*}
\begin{enumerate} 
\item  求前一个基到后一个基的过渡矩阵。
\item  求 $A=\begin{bmatrix} -3&5 \\ 4&2 \end{bmatrix}$ 在后一组基下的坐标。
\end{enumerate} 

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%\begin{enumerate} 
%\item  写出过渡矩阵的定义里的那个等式。
%\item  写出坐标的定义里的那个等式。
%\end{enumerate} 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %40
设 $A, B$ 分别是 $m\times n$ 与 $n\times s$ 矩阵。证明：如果 $AB=0$, 那么 $R(A)+R(B)\leq n$.

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：矩阵 $B$ 的每个列向量都是线性方程组 $AX=0$ 的解向量。
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %41
设 $A$ 是 $n$ $(n\geq 2)$ 阶矩阵, 证明: 
$%\begin{eqnarray*}
R(A^*)=\left\{\begin{array}{ll}
n, & R(A)=n, \\
1, & R(A)=n-1, \\
0, & R(A)<n-1. \\
\end{array}\right.
$%\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：使用下述定理：\\
%(1) 记 $d=\det(A)$, 则有 $AA^*=dE$. \\ 
%(2) $AX=0$ 的解空间的维数是 $n-R(A)$.
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %42
设实向量空间 $V=R^3$ 中的一些向量定义如下: 
\begin{eqnarray*}
\alpha_1=(1,2,0),\alpha_2=(0,1,2),\alpha_3=(2,0,1),\\ 
\beta_1=(1,1,0),\beta_2=(0,1,1),\beta_3=(1,0,1).
\end{eqnarray*}

\begin{enumerate}
\item  证明向量组 $\Phi=\{\alpha_1,\alpha_2,\alpha_3\}$ 是 $V$ 的一个基。
\item  证明向量组 $\Psi=\{\beta_1,\beta_2,\beta_3\}$ 是 $V$ 的一个基。
\item  求从 $\Phi$ 到 $\Psi$ 的过渡矩阵。
\item  设向量 $\xi$ 在 $\Phi$ 下的坐标是 $(1,2,3)$, 求这个向量在 $\Psi$ 下的坐标。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%\begin{enumerate}
%\item  验证行列式的值不为零。
%\item  验证行列式的值不为零。
%\item  写出过渡矩阵的定义里的等式。
%\item  写出坐标的定义里的等式。
%\end{enumerate}
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item %43
设 $\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5$ 是实向量空间 $V$ 中的一个线性无关向量组。 
证明：$W_1=L(\alpha_1, \alpha_2)$ 与 $W_2=L(\alpha_3, \alpha_4, \alpha_5)$ 的交子空间是零子空间。

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：设有 $\xi\in W_1\cap W_2$, 则有实数 $k_1,k_2,k_3,k_4,k_5$ 使得
%\begin{eqnarray*}
%\xi &=& k_1\alpha_1+k_2\alpha_2, \\ 
%\xi &=& k_3\alpha_3+k_4\alpha_4+k_5\alpha_5.
%\end{eqnarray*}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %44
设 $V$ 表示实数域 $\mathbb{R}$ 上次数小于等于 3 的所有多项式组成的向量空间。

\begin{enumerate}
\item  求 $W=L(x^3+1, x^2-x+1, x+1)$ 的维数和一个基。
\item  求 $W$ 的一个余子空间。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%\begin{enumerate}
%\item  建立 $V$ 与 $\mathbb{R}^4$ 的一个同构，求出这个向量组的一个极大线性无关组。
%\item  将上一小题求得的极大线性无关组扩充为 $\mathbb{R}^4$ 的一个基。则加进去的向量线性张成了所求的一个余子空间。
%\end{enumerate}
%
%}

\vspace{0.2cm}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %45
考虑 $V=\mathbb{R}^4$ 中的两个向量 $\alpha_1=(1,2,3,4)$, $\alpha_2=(5,6,7,8)$. 设 $W=L(\alpha_1, \alpha_2)$, 求$W$ 在 $V$ 中的一个余子空间。


%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：求 $\alpha_3,\alpha_4$ 使得向量组 $(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$ 成为 $\mathbb{R}^4$ 的一个基。
%则所求余子空间可以取为 $L(\alpha_3,\alpha_4)$. 
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item  %46
已知 $V=\mathbb{R}^5$ 中的几个向量，
\begin{eqnarray*}
\alpha_1=(1,1,0,0,0), \alpha_2=(0,0,1,1,0),\alpha_3=(0,1,1,0,0),\alpha_4=(0,0,0,1,1), \alpha_5=(1,0,0,1,0).
\end{eqnarray*}
考虑子空间 $W_1=L(\alpha_1, \alpha_2)$ 和子空间 $W_2=L(\alpha_3, \alpha_4, \alpha_5)$.

\begin{enumerate}
\item  求和子空间 $W_1+W_2$ 的一个基和维数。
\item  求交子空间 $W_1\cap W_2$ 的一个基和维数。
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%
%\vspace{0.2cm}
%
%{\color{red}解答：
%\begin{enumerate}
%\item  将向量 $\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5$ 按列向量的方式排成一个矩阵，然后化为行最简形。
%\item  设 $x_1\alpha_1+x_2\alpha_2 = x_3\alpha_3 + x_4\alpha_4+x_5\alpha_5$. 
%将 $(x_1,x_2,x_3,x_4,x_5)$ 看作未知数，求解线性方程组。
%\end{enumerate}
%
%}

\vspace{0.2cm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{enumerate}

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\end{document}
